% GG
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage{dsfont}
\usepackage{amsmath}
\usepackage{epsfig}
\usepackage[usenames,dvipsnames]{color}
\usepackage{graphicx}
\usepackage{url}
\usepackage{multirow}

% Natbib setup for author-year style
\usepackage{natbib}
 \bibpunct[, ]{(}{)}{,}{a}{}{,}%
 \def\bibfont{\small}%
 \def\bibsep{\smallskipamount}%
 \def\bibhang{24pt}%
 \def\newblock{\ }%
 \def\BIBand{and}%

% ganx packages and commands
% \usepackage[includeheadfoot,top=20mm, bottom=20mm, footskip=2.5cm]{geometry}
% \usepackage[includeheadfoot,left=15mm,right=15mm,top=20mm, bottom=20mm, footskip=2.5cm]{geometry}
% https://tex.stackexchange.com/questions/54055/using-lower-case-roman-numerals-in-enumerate-lists
% https://tex.stackexchange.com/a/58853/104838
% \usepackage{enumerate}
\usepackage[shortlabels]{enumitem}
\usepackage{tabularx}
\newcolumntype{Y}{>{\centering\arraybackslash}X}% define horizotal centering Y
\renewcommand\tabularxcolumn[1]{m{#1}}% for vertical centering text in X column

% https://liam.page/2017/05/01/the-correct-way-to-use-differential-operator/
\newcommand*{\dif}{\mathop{}\!\mathrm{d}}
% https://readthelatexmanual.wordpress.com/2016/02/21/roman-numerals-in-latex/
\newcommand{\RomanNumeralCaps}[1]
    {\MakeUppercase{\romannumeral #1}}
\newcommand{\romanNumeral}[1]
    {\Makelowercase{\romannumeral #1}}

% GG commands.
\newtheorem{result}{Result}
\renewcommand{\baselinestretch}{1.5}
\newtheorem{Definition_}{Definition}
\newtheorem{Result_}{Result}
\newtheorem{Theorem_}{Theorem}
\newtheorem{Proposition_}{Proposition}
\newtheorem{Property_}{Property}
\newtheorem{Remark_}{Remark}
\newtheorem{Lemma_}{Lemma}
\newtheorem{Corollary_}{Corollary}
\newtheorem{Example_}{Example}
\newcommand{\rd}{{\rm d}}
\newcommand{\E}{{\mathds{E}}}
\newcommand{\I}{{\mathds{I}}}
\newcommand{\Z}{{\mathds{Z}}}
\newcommand{\R}{{\mathds{R}}}

%\newcommand{\Pr}{{\rm Pr}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\bean}{\begin{eqnarray*}}
\newcommand{\eean}{\end{eqnarray*}}
\newenvironment{pf}{\vspace{0.1in}\noindent \textbf{Proof of}\!\! ~}{~
\hspace*{\fill} $\square$ }
\newenvironment{pfa}{\vspace{0.1in}\noindent \textbf{Proof.} ~}{~
\hspace*{\fill} $\square$ }

% \marginparwidth 0pt\marginparsep 0pt \topskip -0.1  pt\headsep
% 0pt\headheight 0pt \oddsidemargin -0.in\evensidemargin 0pt
% \textwidth 6.5in \topmargin 0pt\textheight 9in
\newcommand{\commentout}[1]{}
